Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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31 views

Poincare' s inequality for vectorfields on the sphere

Let $\mathbb{S}^2$ be the standard 2-sphere, and let $V$ be $\mathcal{C}^1$ vectorfield on it. I'd like to understand if it is true that there exists $C > 0$ such that, for all such $V$, we have $$ ...
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1answer
36 views

Computing distance on a sphere

Let's say I want to compute the distance between two far away points on Earth. For example, let's say I want to compute the distance from Toronto to Brazil. I can do this by getting in my car, setting ...
1
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1answer
24 views

Curvature of the pseudosphere

I have the parametrization $x(u,v)=(\cos u \sin v, \sin u \sin v , \cos v+\log (\tan {v/2}))$ with $0<v<\pi$ y $0<u<2\pi$. From this parametrization, how can I compute (optimally) the ...
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1answer
20 views

Inexistence of periodic orbits using Bendixson's criterion

Let $X:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a linear map. Prove that there is $\delta>0$ such that for all field $Y$ over $\mathbb{R}^2$ satisfying: $$ \underset{x\in\mathbb{R}^2}{\sup} ...
4
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1answer
96 views

Euler characteristic, genus and cohomology: a deep connection?

For a smooth projective curve $V$ over the complex numbers, the algebraic genus, defined as the dimension of the linear system $L(\omega)$, where $\omega$ is the canonical divisor, coincides with the ...
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1answer
27 views

Angular bracket operation in differential geometry?

There is an angular bracket operation in geometry, which looks like $$\langle X,Y\rangle$$ where $X$ and $Y$ are apparently $(0,1)$ tensors. It appears for instance in the answer to the following ...
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1answer
23 views

Poincare type inequality on compact manifold

I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact manifold with or without boundary. The inequality I am looking for is the equivalent of $ ...
2
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1answer
67 views

Source for Differential Manifolds/Geometry Questions?

I'm looking for a good source for a large collection of Differential Manifolds/Geometry questions covering a subset of the following topics: inverse function theorem, local coordinates, induced ...
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0answers
21 views

Electrodynamics and U(1)-gauge model [on hold]

As the article 'Electrodynamics in general spacetime' greatly explains, the U(1)-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to ...
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80 views
+100

Proof of holomorphic Lefschetz fixed point formula using currents in Griffiths and Harris

I am trying to understand the proof of the Holomorphic Lefschetz fixed point formula on page 426 in Griffiths and Harris. However, I find their use of currents extremely confusing. They seem to go ...
0
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1answer
31 views

Definition of Linear independence of algebraic $1$-forms

"Suppose that $a,b,c,d$ are linearly independent algebraic $1$-forms on $\mathbb{R}^n$". What does it mean for algebraic $1$-forms to be linearly independent? I have looked through my notes and ...
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1answer
42 views

Start of the proof of the Poincaré Lemma

Let $\hat{\Phi}:U \rightarrow U$ be a one-parameter family of diffeomorphisms defined for $\ 0 < t \leq 1$. Let $\beta \in \Omega^k(U) $ be a closed k-form. Suppose that $\hat{\Phi}^{*}_1 \beta = ...
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35 views

Differential Geometry for Computer Science

I am looking for a good book or other resources on Differential Geometry for Computer Sciences or more specifically Differential Geometry used in Computer Graphics, Geometric Modelling and Mesh ...
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0answers
18 views

fibration sequence of projective spaces

Question~1: How to construct a fibration sequence $$ S^3\to S^2 \to \mathbb{C}P^\infty\to \mathbb{H}P^\infty ? $$ Does $$S^3\simeq \Omega \mathbb{H}P^\infty ? $$ (Since $\mathbb{C}P^\infty\simeq ...
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1answer
36 views

tangent space of manifold and Kernel

I want to understand the proof of this Theorem : Theorem: Let $ S $ be a submanifold of $ E $ and let $ a \in S $. Assume that there exists a submersion $f : E \supset U \rightarrow F$ of class $ ...
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2answers
34 views

Lie Group Automorphism which are diffeomorphism

Is every smooth automorphism of a Lie Group $G$ a diffeomorphism?
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0answers
36 views

Why not differentiable manifolds that are not of class $C^1$

In most, if not all (I cannot say for sure) references on manifolds, we seem to consider $C^k$-manifolds, including the case $k = 0$, which corresponds to topological manifolds. This means that we ...
2
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2answers
37 views

Is it possible to build a fiber bundle whose fibers are different? (Or we should not call it a fiber bundle?)

Suppose there is a fiber bundle $E$. The base space is $M$ so that $\pi:E\rightarrow M$ is the projection. By the definition, the bundle has a typical fiber $F$ such that the local trivialization over ...
3
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1answer
34 views

Turning number VS winding number

To avoid confusion, here are the definitions of the objects in this question: 1) Let $\gamma:S^1\to\mathbb{R}^2\setminus\{0\}$ a smooth loop. The winding number of $\gamma$ is the number of times ...
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0answers
30 views

Gaussian curvature proof

I can show the first part but not sure how to proceed after that.
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0answers
14 views

Surjective $\gamma \colon I \to M^1$, $\gamma (t_1)=\gamma (t_2)$ can be extended to a periodic parametrization of $M^1$

Suppose that $\gamma \colon I \to M^1$ is a smooth surjective curve in a Riemannian connected 1-dimensional manifold. Furthermore, suppose that it is parametrized via arc lenght i.e.:$$||\dot ...
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12 views

Existence of CAT(0)-metrics

Given a metrisable, contractible topological space. Under which conditions exists a CAT(0)-metric compatible with the given topology?
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0answers
20 views

Symmetry of Christoffel symbols of the second kind

I was reading the article: http://physicspages.com/2013/12/22/christoffel-symbols-symmetry/, and I do not understand this: In the locally flat frame, this equation reduces to $\displaystyle ...
2
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1answer
63 views

Submanifold associated to blow up.

I 'm trying to understand the classical blow up given by $$X=\{(x,[y])\in \mathbb{R}^n \times \mathbb{P}_N / \hspace{0.2cm} \exists \lambda \in \mathbb{R} \hspace{0.3cm} \text{such that} ...
0
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1answer
32 views

Local coordinates on a product of two manifolds.

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. I think that a local coordinate on $X \times Y$ is $(U \times V, x_1 ...
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2answers
40 views

Parametrization of a unit 2 sphere

Here is the parametrization for a unit 2 sphere locating at the center of a Euclidean 3 dimensional space: $$x=x(u,v)= \cos u\sin v,\ \ y=y(u,v)=\sin u\sin v,\ \ z=z(u,v)=\cos v, $$ where $0\leq ...
2
votes
4answers
189 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
2
votes
1answer
276 views

Differential forms and a chain rule

Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$. Let $Q\in U$ ...
3
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1answer
49 views

Inner product, differential forms and surfaces (Stokes' theorem)

I'm trying to understand how do you get the Kelvin-Stokes theorem \begin{equation} \int_{S} (\nabla\times \omega) \cdot \mathrm{d}S = \int_{\partial S} \omega \cdot \mathrm{d}r \end{equation} from the ...
3
votes
1answer
41 views

Integration by parts (Differential Geometry)

I am studying the proof of a theorem and I am stuck. It says that by integration by parts we get that: For $g(t)$ a variation of Riemannian metrics wih $g'(0)=h,$ $$\int_{M} (-\Delta (tr h) + ...
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0answers
15 views

For two unit speed curves $\gamma_{1,2}$ on $[0,l]$ where $0\leq k_2\leq k_1<\pi/l$, show $d(\gamma_1(0),\gamma_1(l))\leq d(\gamma_2(0),\gamma_2(l))$.

For two unit speed curves $\gamma_{1,2}$ on $[0,l]$ where $0\leq k_2\leq k_1<\pi/l$, show $d(\gamma_1(0),\gamma_1(l))\leq d(\gamma_2(0),\gamma_2(l))$. It is important to note here that ...
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0answers
22 views

Definition of a connection on a principal bundle

I am trying to understand the definition of a connection as given in, for example, Taubes' book Differential Geometry. Let $\pi: P \to M$ be a principal bundle with a $G$ action. He states that a ...
1
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1answer
21 views

Counting negative eigenvalues of a Hessian.

Let $f:M\to\Bbb{R}$ be a Morse function. The number of negative eigenvalues of the Hessian at a non-degenerate critical point is the index of $f$ at that critical point. When counting negative ...
1
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1answer
44 views

Differential geometry; evaluating the differential $df$ of a function $f$ from the sphere to a meridian and the first fundamental form

Let $C$ be the meridian $C= \{ (x,y,z) \in \Bbb S^2 | y=0,x\geq 0 \}$. Let $f$ map the sphere $\Bbb S^2$ to $C$ such that $f$ maps every point on the sphere to the unique point on $C$ with the same ...
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0answers
16 views

Meaning of a Hypersurface resulting from Lagrange Multipliers

Suppose we have a function $f(x_1,\ldots,x_n)$ that we wish to maximize under the set of $n-1$ constrictions $g_i(x_1,\ldots,x_n) = c_i$ for $i \in \{1,\ldots,n-1\}$. We write the Lagrangian ...
2
votes
1answer
49 views

Solvability of system of differential equations

Given $a_i:\mathbb{R}^n \to \mathbb{R}$ $(1\leq i \leq n)$, I am trying to find the conditions under which the equations $$ \frac{\partial f}{\partial x^i}=a_i(x_1,...,x_n) $$ $$ f(x_0)=z_0 $$ is ...
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0answers
24 views

On the Euler characteristic in Morse Theory

Let $f:M\to \Bbb{R}$ be a Morse function, where $M$ is a $k$-manifold. The index $i_{f,p}$ is defined to be the number of negative eigenvalues of the Hessian $H_f$ at the critical point $p$. For ...
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0answers
34 views

Stuck in Preissmann's theorem

I am stuck on following the proof of Preissmann's theorem, whose statement is that Let $(M,g)$ be a closed connected Riemannian manifold of negative sectional curvature. Then every nontrivial ...
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0answers
32 views

Angle of intersection between a plane and sphere.

Let $X(\theta,\phi)=(\sin \theta \cos \phi, \sin\theta\sin \phi, \cos\theta)$ be parametrization of the sphere $S^2$. Let $P$ be the plane $x=z \cot\alpha$, $0<\alpha<\pi$ and $\beta$ be the ...
2
votes
1answer
14 views

Singular chain complex for integration - pinching on boundary

Singular chain complex, as far as topology are concerned, is just continuous map from standard simplex, and the choice of using simplex over other shape is immaterial. But for integration on manifold, ...
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1answer
31 views

When is a vector field on a manifold restricted to a submanifold $X$ a vector field on $X$?

Let $X$ be an embedded submanifold of $M$ and let $V$ be a vector field on $M$. One can restrict $V$ to $X$, but it may not define a vector field on $X$. Example: The vector field $x^i\partial_i$ on ...
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21 views

Find the lie derivative of a particular integral

Let $\mathbb{X}=(1,y)$ be a vector field on $\mathbb{R}^2$. Let $\Phi_t$ be the flow of $\mathbb{X}$. The flow of $\mathbb{X}$ I have calculated to be $\Phi_t(x,y)=(x+t,ye^t)$ Given a function ...
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0answers
11 views

Poisson bivector on the product of two manifolds

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. A Poisson bivector on $X$ is defined by \begin{align} \pi_X = \sum_{i,j} ...
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38 views

Application Question - American universities strong in Differential Geometry?

Can anyone recommend some American universities (except those top 10 ones such as Harvard, Princeton, SUNY and Umichgan etc. ) which have departments with a solid focus on Geometry and Topology, ...
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33 views

Are there some reference books or handbooks on homology and homotopy groups of every manifold which has been calculated?

Are there some reference books or handbooks on homology and homotopy groups of every manifold which has been calculated ? Or Are there some reference books especially on differential geometry and ...
3
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1answer
36 views

If $\operatorname{Def} X = \frac{1}{2} \mathcal L_X g$ then what is $\operatorname{Def}^* \operatorname{Def} X$?

Let us define a deformation operator $\operatorname{Def}$ on a Riemannian manifold $(M,g)$, acting on divergence-free vector fields as $$ \operatorname{Def} X = \frac{1}{2} \mathcal L_X g, $$ where ...
1
vote
1answer
141 views

Extension of a map $g:\overline{B_1^n}\to \mathbb{R}^2$

Let $B_r^n\subset \mathbb{R}^n$ ($n\geq 6$) be the open ball with radius $r$ and let $g:B_2^n\to \mathbb{R}^2$ be an analytic map. How to define a continuous map $h:\mathbb{R}^n\to \mathbb{R}^2$ such ...
1
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1answer
59 views

A difficult question on mathematical physics

Let $TQ^*$ be equipped with its standard symplectic structure and let $X_H$ be a Hamiltonian vector field which is tangent to the fibers of $\pi: TQ^* \to Q.$ I need to show that $$H=h \circ \pi = \pi ...
10
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2answers
201 views

Are compact complete geodesics closed?

Let $(M,g)$ be a compact Riemannian manifold. Is there an example of a geodesic $c:\mathbb{R}\to M$ s.t. $c(\mathbb{R})$ is compact, $c$ is NOT periodic (i.e. be NOT a closed geodesic) ?
2
votes
0answers
29 views

Parameterization of surface of revolution with constant mean curvature

Let $x(u,v) = (g(u), h(u) \cos v, h(u) \sin v)$ be a parameterization of a surface of revolution $M$, arising from rotating the regular curve $\alpha(u) = (g(u),h(u),0)$ around the $x$-axis with ...